Victor Dickenson
6161c
4/4/04
Beware the Matrix
This Learning Activity Packet (LAP) introduces adding, subtracting and multiplying simple matrices. The matrices have been constructed so that no integers are used and the students will use basic addition, subtraction and multiplication skills to solve the problems. The algorithm for multiplying a matrix by a matrix, however, may be a tricky pattern for students to get at first. The activities may be modified to introduce rational numbers and integers. The difficulty of the LAP is middle grade level.
Beware the Matrix
Although the title of this activity says beware the matrix, you will find this activity challenging but not impossible. You will use the addition, subtraction, and multiplication skills that you already know to help you work with matrices.
Let’s first learn how to add matrices. You can only add one matrix to another matrix if the matrices are the same size. Let’s first work with 2 by 2 matrices. The numbers in a matrix are called elements. We will use e1, e2, e3, and e4 for the elements in one matrix and E1, E2, E3 and E4 for the elements in another matrix. We add e1+E1, e2+E2, e3+E3 and e4+E4 and place them in a matrix.
+ =
Now with numbers:
+ = which is
Not too bad! Remember to add the first elements together and the second elements together and so forth.
Now it is you turn
+ = + =
Did you get for the first answer and for the second answer?
Lets try a few more examples, what happens when we add + is it the same as + ?
+ = =
+ = =
Yes they are the same. Just like 5+3 = 3+5, when we add matrices together it doesn’t matter which order we add them the sum is still the same. This is called the commutative property of addition.
Exercise 1
Make up your own 2 by 2 matrix. Find the sum of your matrix and .
Make up another 2 by 2 matrix; once again add to your matrix. Do you see a pattern? Make up a rule for adding any 2 by 2 matrix with . The matrix is called the zero matrix.
Challenge 1
Finish the sum of these 3 by 3 matrices.
+ = =
What would the sum be? + =
Subtraction of matrices is similar to addition of matrices. For example:
- = =
- = =
Exercise 2
What is the solution? - =
Challenge 2
What matrix solves the equation?
- =
You can also multiply a matrix by a single number, this is called scalar multiplication. For example:
multiplied by the scalar 4 = =
and multiplied by the scalar 5 = =
Exercise 3
Solve these: multiplied by the scalar 7 =
multiplied by the scalar 4 =
What do you think any matrix multiplied by the scalar zero would be?
Make up a matrix and multiply it by the scalar zero to find out.
x =
Let’s look at this pattern or algorithm using numbers.
x = = =
You know that 5 x 1 = 5 and 12 x 1 = 12. You know that any number multiplied by one is equal to the number itself. One is called the multiplicative identity. What would happen if we multiply and , do you think we will get ? Let’s try it.
x = = =
You see is not the same thing as !
Exercise 4
Finish this problem x = =
Solve these problems x =
x =
What do you think that times any 2 by 2 matrix is equal to?
Our last investigation of matrices will be multiplying a 2 by 2 matrix by a 2 by 2 matrix. The pattern is a little tricky, but if you are careful you will have it in no time. Let a, b, c, d, e, f, g, and h be numbers.
x =
And now with numbers
x = = =
Exercise 5
Finish this problem
x = =
x = =
Another interesting property of matrices is that multiplication of matrices is not commutative. You know that 5 x 2 = 2 x 5 and 9 x 6 = 6 x 9,in fact, any a x b = b x a for any whole number a and b. This is called the commutative property of multiplication and shows that order does not matter when multiplying whole numbers. Above exercise 5 we showed that:
x = = =
But does
x = ?
x = = =
So clearly the two problems are not equal. In general, the order in which you multiply matrices does matter because changing the order usually results in different answers.
We say multiplying a matrix by a matrix is generally not commutative.
Challenges 3-6
3. The matrix is called the identity matrix for all 2 by 2 matrices. Multiplying any original 2 by 2 matrix by the identity matrix will give you the original matrix. Make up a 2 by 2 matrix and multiply it by to show that the answer will be the matrix that you made up.
4. Make up another 2 by 2 matrix and multiply it by the zero matrix . What was you answer? Can you come up with a rule for multiplying a 2 by 2 matrix with ?
5. What happens when you multiply a 2 by 2 matrix by ? What about multiplying by ?
6. You know that if you multiply any number by zero you get zero. What happens when you multiply by itself?
Answer key
Exercise one AWV when the student adds their matrix to the zero matrix the sum should be the same as their original matrix.
Challenge onea) b)
Exercise 2
Challenge two
Exercise 3 a) b)
Exercise 4 a) b) c)
Exercise 5 a) b)
Challenge 3 AWV however the student should end up with their original matrix after
multiplying it with the identity matrix
Challenge 4 a. b. any 2 by 2 matrix multiplied by the zero matrix is
Challenge 5 AWV When you multiply a matrix by all the entries or elements
double.
When you multiply a matrix by all the entries or elements
triple.
Challenge 6 x = . The matrix has a nilpotency of 2, which
means when you square the matrix it yields the zero matrix.